Is it possible to allow functional updates "{< ... >}" in object initializers (or another keyword for such cases)? I.e., allow initializers to return objects of self type? Initializers are quite useful for sharing code between constructors and methods, but it can't be used with functional objects (without making instance variables mutable).

You raise an interesting question.
The main difficulty I see is that it would require a change in initializer semantics:
currently an initializer receives self and returns nothing, and you suggest that it should
return a new self. It seems hard to make this compatible with the current behaviour,
except by discriminating on the return type in a dirty way.

Note that you can already change the internal state of an object when inheriting, just
by overwriting it. What is missing is the ability to access the previous state. It could be
possible, because the system already protects against accessing variables when a value field
with the same name is defined.
Consider this program:

Currently it is refused:
Error: The instance variable x
cannot be accessed from the definition of another instance variable

Allowing it would mean that the field definition in c overrides the x from
d's constructor. I've heard that this kind of situation is allowed in Scala, but
very confusing, so I'm not sure allowing it would be a good idea...

Allowing instance variable definitions to refer to previous definitions doesn't quite cover the use case I described - sharing code in methods - since the definition cannot refer to self (unless self is treated as an instance variable too?). To give a more concrete example, a set class might be defined as:

Furthermore, initializer currently allow mutable objects to be updated, which can already lead to the situation you describe (depending on the order of inherit/initializer). While it may potentially be confusing, but it's also quite convenient. It would be nice to give functional objects similar capabilities.